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2 Answers
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In Magic, when one event triggers multiple abilities simultaneously, you put all the triggers on the stack and resolve them one by one, sequentially.

So, what's actually happening with the triggers is this:

Both Plague Spitters' abilities trigger at the same time. Since you control both, you can put them on the stack in whatever order you choose. We'll say you put ability #1 on the stack first, then ability #2.

Stack is: PS1, PS2.

If nobody adds more stuff to the stack, we begin resolving abilities on the stack. Plague Spitter ability #2 at the top of the stack resolves and does 1 damage to each creature. You check to see if any creatures died, but they all have 2 toughness, so they don't. Also, the damage triggers Rite of Passage. You put the Rite of Passage triggers (one for each creature) on top of the stack. If there are three creatures, there are three separate triggers.

Stack is: PS1, RoP1, RoP2, RoP3.

(Note that the RoP triggers are on the stack, so none of your guys have +1/+1 counters yet, but they all do have one damage marked on them. If I Gut Shot one of your Plague Spitters right now, I could wipe your board using its death trigger.)

All pass. We resolve RoP trigger #3.

Stack is: PS1, RoP1, RoP2.

All pass. We resolve RoP trigger #2.

Stack is: PS1, RoP1.

All pass. We resolve RoP trigger #1. (Now all your creatures have a +1/+1 counter on them.)

Stack is: PS1.

All pass. We resolve Plague Spitter trigger #1. The Plague Spitter does 1 damage to each creature. We check if any of them died. They all have three toughness now, because of the counters, so nope. Rite of Passage triggers again (for each creature).

Stack is: RoP4, RoP5, RoP6.

(Continue resolving RoP triggers.)

You'll end up with a bunch of 4/4 creatures (2/2s with two +1/+1 counters each) with 2 damage marked on them.

So this happens because I control both Plague Spitters. What if my opponent was in control of one of the Plague Spitters? would that make any difference regarding stack sequence?
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Jacob LundFeb 18 '14 at 20:38

@JacobLund You go in "APNAP" order, which means the active player puts all of his or her triggers on the stack first in any order, then everyone else goes in turn order. (Meaning that your opponent's triggers will go on the stack above yours, and thereby resolve first.) See 603.3b and 101.4 in the comp rules.
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Alex PFeb 18 '14 at 20:42

@AlexP Yeah, but since both PS trigger before anything else, the order is actually rather irrelevant in this case, but their PS will die to the second trigger, as it didn't get buffed by RoP.
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CruncherFeb 20 '14 at 19:38

Your creatures will live, the Plague Spitter's abilities resolve separately. At the end of your upkeep you will have 3 4/4 creatures that are marked with 2 damage each (2 plague spitters and your starting 2/2 creature.